JANUABT 20, 1911] 



SCIENCE 



85 



librating, the upper limit of the eccentri- 

 city is at least one fifth. Thus at some time 

 the asteroid must have a comparatively 

 large eccentricity. These large changes in 

 the eccentricity take place very slowly. 

 Hence the asteroid will during one interval 

 perform many revolutions round the sun 

 in a nearly circular orbit and during 

 another interval many revolutions in a 

 quite eccentric orbit. 



It has been mentioned that these large 

 variations of the eccentricity are of very 

 long period — the same as that of the 

 libration — and that they increase with the 

 extent of the libration. One may regard 

 the asteroid as slowly extracting energy 

 from Jupiter (or losing it to Jupiter) until 

 it has accumulated (or lost) a maximum 

 amount, when the reverse process takes 

 place. Jupiter, owing to its immense mass 

 in comparison with that of the asteroid, 

 will not have its motion sensibly altered by 

 the loss or gain. If the type of motion 

 changes when this loss or gain is near a 

 maximum, the asteroid may in future so 

 move that the energy never returns to its 

 original source. The type must then be 

 unstable. 



In my discussion all powers of the eccen- 

 tricity but the lowest have been neglected. 

 This omission prevents us from obtaining 

 a proper representation of the motion 

 when the eccentricity has risen to so large 

 a value as one fifth, and it is to be remem- 

 bered that this is the lowest maximum 

 which any librating asteroid can have. 



If we take into account higher powers of 

 the eccentricity, it is seen that the extent 

 of the variation must be increased to a con- 

 siderable extent. I have not so far been 

 able to obtain numerical results beyond 

 this stage. The indications are that the 

 eccentricity increases so much at some time 

 that the stability of this type of motion is 

 threatened and that the asteroid is com- 



pelled to take up some other type which is 

 more stable. 



Although the facts I have stated do not 

 furnish a proof that librations round the 

 ratio 2 : 1 can not exist, they at least indi- 

 cate the direction in which stability be- 

 comes doubtful. The usual analytical rep- 

 resentation, which is possible for moderate 

 librations of the period, fails where the ec- 

 centricity becomes very small or great, and 

 the failure is partly due to the short period 

 terms. The question which requires solu- 

 tion is whether the failure with large ec- 

 centricities is merely a defect of the mode 

 of representation, or whether it is due to 

 actual instability. My own impression is 

 that the limits of stability of libration 

 about the principal ratios are so near to- 

 gether, that the chance of an asteroid 

 lying between them is very small. This 

 impression is, however, not independent of 

 the observations. There are no known 

 asteroids which perform librations round 

 the ratio 2 : 1 and there are none which ap- 

 proach libration very closely. The lacuna 

 in the distribution doubtless admits of 

 other interpretations, but the framing of 

 hypotheses to explain it must properly 

 await a more complete discussion of the 

 consequences of the law of gravitation. 



The distribution of the asteroids on 

 either side of the gap furnishes certain 

 tests. The theory indicates that asteroids 

 which initially had large eccentricities are 

 more likely to librate than those with small 

 eccentricities. If the large eccentricities, 

 as I believe, tend to become unstable, it fol- 

 lows that the gap must be wider for them 

 than for the small eccentricities. The next 

 slide (Fig. 2) shows the distribution of 

 eccentricities greater and less than one 

 fifth, round the ratio 2:1. It shows also 

 the distribution by differences of .05 of the 

 eccentricities above .20. The tendency of 

 the asteroids to be more distant from the 



